XVA Pricing A. Capponi Arbitrage-Free Pricing of XVA Motivation Model Hedging Agostino Capponi Arbitrage Columbia University Theory Explicit Examples joint work with Maxim Bichuch (WPI) and Stephan Sturm (WPI) PDE Repre- sentations Conclusion IAQF/Thalesians Seminar Series New York, September 21, 2015
The LIBOR-OIS Spread XVA Pricing A. Capponi Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion
The LIBOR-OIS Spread XVA Pricing A. Capponi Consequences Motivation Widening of spreads is due to counterparty credit risk Model LIBOR cannot be considered a risk-free rate any longer Hedging One cannot assume the existence of a universal risk-free Arbitrage rate r Theory Explicit Examples PDE Repre- sentations Conclusion
The LIBOR-OIS Spread XVA Pricing A. Capponi Consequences Motivation Widening of spreads is due to counterparty credit risk Model LIBOR cannot be considered a risk-free rate any longer Hedging One cannot assume the existence of a universal risk-free Arbitrage rate r Theory Explicit Examples Rates at which derivatives traders borrow and lend PDE Repre- unsecured cash differ sentations How to price and hedge derivatives in presence of funding Conclusion spread and counterparty risk?
The LIBOR-OIS Spread XVA Pricing A. Capponi Consequences Motivation Widening of spreads is due to counterparty credit risk Model LIBOR cannot be considered a risk-free rate any longer Hedging One cannot assume the existence of a universal risk-free Arbitrage rate r Theory Explicit Examples Rates at which derivatives traders borrow and lend PDE Repre- unsecured cash differ sentations How to price and hedge derivatives in presence of funding Conclusion spread and counterparty risk? 2013: Many banks (Barclays, JPM, BoA,...) introduce XVA desks
Literature XVA Pricing A. Capponi Motivation Practitioner literature: Piterbarg (2010, 2012), Burgard & Model Kjaer (2010, 2011), Mercurio (2013) Hedging Arbitrage Theory (Corporate) Finance literature: Hull & White (2012, 2013) Explicit Examples PDE Repre- sentations Financial Mathematics literature: Bielecki & Rutkowski Conclusion (2013), Brigo (2014), Cr´ epey (2011, 2013), Cr´ epey, Bielecki and Brigo (2014)
Main Contributions XVA Pricing A. Capponi Develop a framework to characterize the total valuation Motivation adjustment (XVA) of a European style claim on a stock in Model presence of Hedging counterparty credit risk Arbitrage Theory funding spread Explicit Examples PDE Repre- sentations Conclusion
Main Contributions XVA Pricing A. Capponi Develop a framework to characterize the total valuation Motivation adjustment (XVA) of a European style claim on a stock in Model presence of Hedging counterparty credit risk Arbitrage Theory funding spread Explicit Examples Derive a nonlinear backward stochastic differential PDE Repre- equation (BSDE) associated with the replicating portfolios sentations of long and short positions in the claim. Conclusion
Main Contributions XVA Pricing A. Capponi Develop a framework to characterize the total valuation Motivation adjustment (XVA) of a European style claim on a stock in Model presence of Hedging counterparty credit risk Arbitrage Theory funding spread Explicit Examples Derive a nonlinear backward stochastic differential PDE Repre- equation (BSDE) associated with the replicating portfolios sentations of long and short positions in the claim. Conclusion Develop an explicit representation of XVA in case of symmetric rates, but in presence of counterparty risk
The Market Model XVA Pricing A. Capponi The market model (I) Motivation Model Treasury desk : borrowing and lending at rates r ✁ f , r � f , Hedging respectively Arbitrage Theory Stock ( S t ): used to the hedge market risk of transaction. Explicit Examples Trading happens through repo market at rates r ✁ r , r � r PDE Repre- (Duffie (1996)) sentations Conclusion Risky bonds ( P I t , P C t ): underwritten by investor/counterparty and used to hedge default risk. Trading does not happen in the repo market
Stock Short-Selling XVA Pricing A. Capponi Stock Market Motivation Model (5) (4) Hedging Arbitrage (1) (2) Theory Explicit Examples Treasury Desk Repo Market Trader PDE Repre- sentations (6) (3) Conclusion r � r Figure: Security driven repo activity: Solid lines are purchases/sales, dashed lines borrowing/lending, dotted lines interest due; blue lines are cash, red lines are stock.
Stock Purchasing XVA Pricing A. Capponi Stock Market Motivation Model (2) (3) Hedging Arbitrage (1) (4) Theory Explicit Examples Treasury Desk Repo Market Trader PDE Repre- sentations (6) (5) Conclusion r ✁ r Figure: Cash driven repo activity: Solid lines are purchases/sales, dashed lines borrowing/lending, dotted lines interest due; blue lines are cash, red lines are stock.
The Market Model XVA Pricing A. Capponi The market model (II) Motivation We consider the dynamics Model Hedging dS t ✏ µ S t dt � σ S t dW t Arbitrage Theory dP I t ✏ µ I P I t dt ✁ P I t ✁ d 1 l t τ I ↕ t ✉ Explicit ✏ ♣ µ I ✁ h I q P I t dt ✁ P I t ✁ d ̟ I Examples t PDE Repre- dP C t ✏ µ C P C t dt ✁ P C t ✁ d 1 l t τ C ↕ t ✉ sentations ✏ ♣ µ C ✁ h C q P C t dt ✁ P C t ✁ d ̟ C Conclusion t for independent default times τ I , τ C with constant default intensities h I , h C and martingales ̟ I , ̟ C
� ↕ � ➔ � ➔ � ↕ ✁ ✁ � ↕ � ↕ ✁ The Market Model XVA Pricing A. Capponi The market model (III) Motivation Can we guarantee that there are no arbitrage opportunities Model in the market? Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion
� ↕ � ➔ � ➔ � ↕ ✁ ✁ � ↕ � ↕ ✁ The Market Model XVA Pricing A. Capponi The market model (III) Motivation Can we guarantee that there are no arbitrage opportunities Model in the market? Hedging Arbitrage Theory As we only model from the point of the trader, we can only conclude this from her perspective. . . Explicit Examples PDE Repre- sentations Conclusion
The Market Model XVA Pricing A. Capponi The market model (III) Motivation Can we guarantee that there are no arbitrage opportunities Model in the market? Hedging Arbitrage Theory As we only model from the point of the trader, we can only conclude this from her perspective. . . Explicit Examples PDE Repre- sentations Proposition Conclusion No-arbitrage conditions: r ↕ r ✁ f , r � ↕ r ✁ f , r � ➔ µ I , r � Necessary : r � ➔ µ C . f f f r ↕ r � Sufficient : Necessary plus r � ↕ r ✁ r f
� ↕ � ➔ � ➔ � ↕ ✁ ✁ � ↕ � ↕ ✁ The Market Model XVA Pricing A. Capponi The market model (III) Motivation Can we guarantee that there are no arbitrage opportunities Model in the market? Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion
� ↕ � ➔ � ➔ � ↕ ✁ ✁ � ↕ � ↕ ✁ The Market Model XVA Pricing A. Capponi The market model (III) Motivation Can we guarantee that there are no arbitrage opportunities Model in the market? Hedging Arbitrage Theory As we only model from the point of the trader, we can only conclude this from her perspective. . . Explicit Examples PDE Repre- sentations Conclusion
The Market Model XVA Pricing A. Capponi The market model (III) Motivation Can we guarantee that there are no arbitrage opportunities Model in the market? Hedging Arbitrage Theory As we only model from the point of the trader, we can only conclude this from her perspective. . . Explicit Examples PDE Repre- sentations Proposition Conclusion No-arbitrage conditions: r ↕ r ✁ f , r � ↕ r ✁ f , r � ➔ µ I , r � Necessary : r � ➔ µ C . f f f r ↕ r � Sufficient : Necessary plus r � ↕ r ✁ r f
� ✁ Collateralization XVA Pricing A. Capponi Motivation Model Collateral is used to secure the derivatives deal Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion
� ✁ Collateralization XVA Pricing A. Capponi Motivation Model Collateral is used to secure the derivatives deal Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion
� ✁ Collateralization XVA Pricing A. Capponi Motivation Model Collateral is used to secure the derivatives deal Hedging Collateral is provided in form of cash (80%) Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion
� ✁ Collateralization XVA Pricing A. Capponi Motivation Model Collateral is used to secure the derivatives deal Hedging Collateral is provided in form of cash (80%) Arbitrage Theory Collateral can be reinvested (rehypothecated) (96%) Explicit Examples PDE Repre- sentations Conclusion
Collateralization XVA Pricing A. Capponi Motivation Model Collateral is used to secure the derivatives deal Hedging Collateral is provided in form of cash (80%) Arbitrage Theory Collateral can be reinvested (rehypothecated) (96%) Explicit Examples PDE Repre- The collateral provider receives interests at rate r � c . The sentations collateral taker pays interests at rate r ✁ c . Conclusion
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